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Cpre537x Final Project
Digital Communication
Via Multipath Fading Channel
Zhiwei Zeng
Instructor: Dr. Russell
November 2000
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Abstract
Multipath fading is a common phenomenon in wireless signal transmission. When a
signal is transmitted over a radio channel, it is subject to reflection, refraction and
diffraction. The communication environment changes quickly and thus introduces morecomplexities and uncertainties to the channel response. This simulator offers a better
understanding of this phenomenon. In order to observe the effects of multipath fadingchannel on the transmitted signal, a whole digital communication system simulator was
developed. Three kinds of digital communication systems: baseband transmission via
additive white Gaussian noise (AWGN) channel, passband transmission via single
AWGN channel, and passband transmission via multipath fading channel, are simulated.
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Contents
Introduction
Baseband transmission via AWGN channel
Signal sourceSampling
Waveform encoding and decoding
Pulse Code Modulation (PCM)
Simple DPCM
General DPCM
Delta ModulationError control coding
Block codesConvolutional codes
Line codingBaseband filtering
Eye pattern
Optimum receiver in AWGN channel
Lowpass filtering
Passband transmission via single AWGN channel
M-ASKQASKM-FSK
M-PSK
Passband transmission via multipath fading channel
Time variant multipath fading channel
Characterization of multipath fading channel
Channel model for multipath fading channel
Signal design for multipath fading channel
RAKE demodulator
Simulation results
Reference
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22.1
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1. Introduction
This report describes a simulator for digital communication systems. Three kinds of
digital communication systems: baseband transmission via additive white Gaussian noise(AWGN) channel, passband transmission via single AWGN channel, and passband
transmission via multipath fading channel, are simulated, as shown in Figure 1. Passbandtransmission via multipath fading channel is emphasized.
Multipath fading is a common phenomenon in wireless signal transmission. When a
signal is transmitted over a radio channel, it is subject to reflection, refraction and
diffraction. Especially in the urban and suburban areas where cellular phones are most
often used, the communication environment changes quickly and thus introduces morecomplexities and uncertainties to the channel response. This simulator offers a better
understanding of this phenomenon.
In order to observe the effects of multipath fading channel on the transmitted signal, awhole digital communication system simulator was developed. For simplicity, complex
techniques, such as multiplexing and spread spectrum communication, were not
simulated. Analyses of the performances of the systems are not given. Section 2 briefly
describes baseband transmission via AWGN channel. Section 3 briefly presents passband
transmission via single AWGN channel. Passband transmission via multipath fading
channel is described in detail in Section 4.
2. Baseband transmission via AWGN channel
Baseband transmission is usually used in wired communication system, such as wired
telephone and wired local area network(LAN). Figure 2 shows the structure of a digital
Fig 1 Contents of the simulator
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communication system with baseband transmission, which consists of signal source,
sampler, waveform encoder, error control encoder, code transformer, baseband filter,AWGN channel, optimum receiver, inverse code transformer, error control decoder, and
lowpass filter. The following subsections describe the above blocks. Users can display
the required figures at different steps in Figure 2 by clicking the small rectangles, whosecolor stands for a display format specified in the legend at the bottom of Figure 2.
2.1 Signal source
A signal source is used to transform information such as voice into electric signal. In
this simulator, a sine waveform is generated with the amplitude, frequency, and initial
phase specified by the user. Figure 3 shows the pre-filtered signal to be transmitted with
amplitude 3V, frequency 1000Hz, and initial phase 45. After some processing, thissignal will be transmitted through an AWGN channel, and be recovered by the receiver.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10-3
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Original signal
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Sampled s ignal
Fig 2 Baseband transmission via AWGN channel
Fig 3 Original signal Fig 4 Sampled signal
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2.2 Sampling
Figure 3 is an analog signal. To be transmitted via a digital communication system,
the signal must be discretized both in time and amplitude. A sampler is to discretize theanalog signal in the time domain. According to the Nyquist sampling theorem, in order to
restore the signal at the receiving site, the sampling frequency should be at least twice ofthat of the maximum signal frequency. Figure 4 shows a sampled signal (discrete signal)
with the sampling frequency 50 times of the signal frequency.
2.3 Waveform encoding and decoding
Waveform encoding converts a source signal into a digital code using a quantizationmethod. The waveform coded signal is represented by a set of integers {1, 2, , N},where N is finite. Waveform decoding recovers the original information signal sequence
using the waveform-coded signal. This simulator supports four waveform-coding
schemes.
2.3.1 Pulse Code Modulation (PCM)
PCM is the simplest and oldest waveform-coding scheme. PCM is a process that
assigns a signal value to inputs that are within a specified range. Inputs that fall in a
different range of values are assigned a different signal value. The input signal is in effect
digitized by scalar quantization, as show in Figure 5. If the quantizer performs uniformquantization, the PCM is called uniform PCM, as Figure 5.
As long as the statistics of the input signal are close to the uniform distribution,
uniform PCM works fine. However, in coding of certain signals such as speech, the input
distribution is far from being uniformly distributed. For a speech waveform, in particular,
there exists a higher probability for smaller amplitudes and lower probability for large
amplitudes. Therefore, it makes sense to design a quantizer with more quantization
regions at lower amplitudes and less quantization regions at larger amplitudes. The
resulting quantizer will be a non-uniform quantizer having quantization regions ofvarious sizes.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10-3
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Signal after waveform decoding
Fig 5 PCM with uniform quantization
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The usual method for performing non-uniform quantization is to first pass the
samples through a nonlinear element that compresses the large amplitudes (reducesdynamic range of the signal) and then perform a uniform quantization on the output. At
the receiving end, the inverse (expansion) of this nonlinear operation is applied to obtain
the sampled value. This technique is called companding (compressing-expanding). Thereare two types of companders that are widely used for speech coding: A-law compander
and -law compander. Figures 6 and 7 are the characteristics of A-law compander and -law compander, respectively. Figures 8 and 9 are the recovered signal using A-law
compander and -law compander, respectively. Except for the discussion of thewaveform schemes in this subsection, the following results are based on PCM with
uniform quantization.
2.3.2 Simple DPCM
When a band-limited random process is sampled at the Nyquist rate or faster, the
sampled values are usually correlated random variables. This means that the previoussamples give some information about the next sample, and this information can be
employed to improve the performance of the PCM system. In the simplest form of
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Signal after waveform decoding
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Signal after waveform decoding
Fig 8 PCM with A-law companding Fig 9 PCM with A-law companding
Fig 6 A-law compander characteristics Fig 7 -law compander characteristics
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differential pulse-code modulation (DPCM), the difference between two adjacent
samples is quantized. Because two adjacent samples are highly correlated, theirdifference has small variations and, therefore, to achieve a certain level of performance
fewer bits are required to quantize it.
Figure 10 shows a block diagram of this simple DPCM scheme. As seen in this
figure, the input to the quantizer is not simply 1 nn XX but rather 1 nn YX . 1 nY is
closely related to Xn-1, and the above choice has the advantage that accumulation of
quantization noise is prevented. Figure 11 shows the recovered signal using simple 4-bit
DPCM system.
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x 10-3
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Signal after waveform decoding
(a) Encoder (b) Decoder
Fig 10 A simple DPCM scheme
Fig 11 Simple DPCM, 4 bits
(a) Encoder (b) Decoder
Fig 12 A general DPCM scheme
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2.3.3 General DPCM
In a more complex version of DPCM, instead of using only the previous sample, the
last p samples are used to predict the value of the next sample. Then the differencebetween the sample Xn and its predicted value is quantized. Usually a linear predictorof
the form = p
i iniXa
1is employed, and the coefficients of the predictor ai are chosen to
minimize the mean-squared error between the sampleXn and its predicted value.
Figure 12 shows a block diagram of a general DPCM system. Because we are using a
p-step predictor, we are using more information in predictingXn and, therefore, the range
of variations ofYn will be less. This in turn means that even lower bit rates are possiblehere. Differential PCM systems find wide applications in speech and image compression.
Figure 13 shows the recovered signal using second-order and 4-bit DPCM.
2.3.4 Delta Modulation
Delta modulation (M) is a simplified version of the simple DPCM scheme. In Mthe quantizer is a 1-bit (2-level) quantizer with magnitudes . Figure 14 shows therecovered signal using M. From Figure 14, we see that when the input signal changesdramatically, the quantized signal cant follow the input signal. This phenomenon is
called overloading. Many schemes have been developed to solve this problem.
2.4 Error control coding
Error-control coding techniques are used to detect and/or correct errors that occur in the
message transmission in a digital communication system. The transmitting side of the
error-control coding adds redundant bits or symbols to the original information signal
sequence. The receiving side of the error-control coding uses these redundant bits of
symbols to detect and/or correct the errors that occurred during transmission. The
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Signal after waveform decoding
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Signal after waveform decoding
Fig 13 General DPCM,p=2, 4 bitsFig 14 M
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transmission coding process is known as encoding, and the receiving coding process is
known as decoding.
2.4.1 Block codes
There are two major classes in error-control code: block and convolutional. In block
coding, successive blocks of K information (message) symbols are formed. The codingalgorithm then transforms each block into a codeword consisting ofN symbols where
N>K. This structure is called an (N, K) code. The ratio K/Nis called the code rate. A key
point is that each codeword is formed independently from other codewords. This
simulator supports three block-coding schemes: Hamming code, cyclic code, and BCHcode.
Figure 15 shows the corrupted signal via an AWGN with variance 0.02 without error-
control coding. Figures 16 through 18 are the signals via the same channel in Figure 15using Hamming code, cyclic code, and BCH code, respectively, withN=7, K=4. In Figure
15, the bit error rate is 0.015. The bit error rates in Figures 16 through 18 are 0s.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
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Signal after waveform decoding
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Signal after waveform decoding
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Signal after waveform decoding
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Signal after waveform decoding
Fig 15 Without error-control coding Fig 16 Hamming code,N=7, K=4
Fig 17 Cyclic code,N=7, K=4 Fig 18 BCH code, N=7, K=4
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2.4.2 Convolutional codes
Convolutional codes differ from block codes in that there are no independent
codewords. The encoding process can be envisioned as a sliding window, Mblock wide,
which moves over the sequence of information symbols in steps of K symbols. M iscalled the constraint length of the code. With each step of the sliding window, the
encoding process generatesNsymbols based on theMKsymbols visible in the window.A convolutional code so constructed is called an (N, K,M) code. Convolutional codes are
commonly used in applications that require relatively good performance with low
implementation cost.
The Viterbi method is used for decoding the convolutional codes. The Viterbi
algorithm is a maximum likelihood(ML) decoding procedure that takes advantage of thefact that a convolutional encoder is afinite state machine. The criterion used for decision
-making is the metric for soft decision decoding and the Hamming distance for hard
decision coding. Details on the Viterbi method can be found in most advanced
communication books.
Figure 19 shows the recovered signal via the same channel as Figure 15 with
convolutional coding. The bit error rate is 0.
2.5 Line coding
Binary 1s and 0s, such as in PCM signaling, may be represented in various serial-bitsignaling formats called line codes. The output of the error-control encoder in this
simulator is unipolar (1s and 0s). Since the direct current component of unipolar signal is
not zero, it is not proper to transmit unipolar signals. Instead, bipolar signaling (1s and
1s) schemes are employed. This simulator provides four bipolar signaling schemes:
bipolar non-return-to-zero signaling (NRZ2), bipolar non-return-to-zero inverse
signaling (NRZI2), bipolar return-to-zero signaling (RZ2), and Millercoding, as shown
in Figure 20. Details of the above signaling schemes can be found in most
communication books. The figures in the following section use Miller code.
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Signal after waveform decoding
Fig 19 Convolutional coding, (3,2,2) code
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6 2 6 4 6 6 6 8 7 0 7 2 7 4- 2
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2.6 Baseband filtering
Rectangular waveforms in Figure 20 have high frequency components. If the channel
bandwidth is limited, rectangular waveforms are not appropriate for this kind of channels.
In this simulator, a raise cosine filteris employed to make the waveform smooth. Figures
3.1 3.2 3.3 3.4 3.5 3.6 3.7
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Signal after baseband filtering
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Fig 20 Line codes
(a) Original (b) NRZ2 (c) NRZI2 (d) RZ2 (e) Miller
Fig 21 Raise cosine filtered signal
for RZ2 signaling
Fig 22 Raise cosine filtered signal
for Miller coding
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1 1.5 2 2.5 3 3.5
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Eye pattern of the signal after baseband filtering
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Eye pattern of the s ignal corrupted by the AW GN
21 and 22 show the raise cosine filtered signals form RZ2 signaling and Miller codes,
respectively.
2.7 Eye pattern
An eye-pattern plot is a simple and convenient tool to study the effects of inter-
symbol interference (ISI) and other channel impairments for digital transmission. The
received signal is plotted against time. When the x-axis time limit is reached, the signalgoes back to the beginning of the time point. Thus the plots overlay each other. In the
optimal condition, the decision pointis at the widest eye opening point. Figures 23 and
24 are the eye patterns of the output of the baseband filtering and the signal corrupted by
the AWGN channel, whose variance is 0.01.
2.8 Optimum receiver in AWGN channel
It is convenient to subdivide the receiver into two parts, the signal demodulator and
the detector. The function of the signal demodulator is to convert the received waveform
r(t) into an N-dimensional vector r=(r1, r2, , rN), where N is the dimension of thetransmitted signal waveforms. The function of the detector is to decide which of the M
possible signal waveforms was transmitted based on observation of the rector r.
The signal and the noise are expanded into a series of linearly weighted orthonormal
basis functions {n(t)}. Two realizations of the signal demodulator are simulated. One isbased on the use of signal correlators, named correlation-type demodulator, as shown in
Figure 25. The second is based on the use ofmatched filters (MF), named matched-filter-type demodulator, as shown in Figure 26. In this simulator, the signal is one-dimensional,
and the realizations of above two demodulators are the same.
The maximum a posteriori probability (MAP) criterion is used to maximize the
probability of a correct decision and, hence, minimizes the probability of error. This
Fig 23 Eye pattern of thebaseband filtered signal
Fig 24 Eye pattern of signalcorrupted by the AWGN channel
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3.1 3.2 3.3 3.4 3.5 3.6 3.7
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Signal corrupted by the AW GN
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Output of the optimum receiver
detector is optimum when the signal has no memory. When the transmitted signal has
memory, the optimum detector is a maximum-likelihood (ML) sequence detector, whichbases its decisions on observation of a sequence of received signals over successive
signal intervals. Although this simulator does not provide ML sequence detector, it has
been shown that a detector based on the MAP criterion and one that is based on the MLcriterion make the same decisions as long as the a priori probabilities are all equal. Figure
27 is the signal, corresponding to that in Figure 22, corrupted by the AWGN channel.Figure 28 is the decisions made by the optimum receiver.
2.9 Lowpass filtering
In order to recover the analog signal from the output of the waveform decoder, a
lowpass filter is needed. This simulator provides a butterworth filterwith the order anddigital cutoff frequency specified by the user. Figure 29 shows the lowpass filtered signal,
Fig 27 Corrupted signal of Figure 22Fig 28 Output of the optimum
receiver for Figure 25
Fig 25 Correlation-type demodulator Fig 26 Matched-filter-type demodulator
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
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Restored s ignal
Original signal
Restored s ignal
with order of 4 and cutoff frequency of 0.1, compared with the original analog signal. It
is obvious that there is a delay due to the infinite impulse response(IIR) filter.
3. Passband transmission via single AWGN channel
Wireless communication systems must use passband transmission. Figure 30 shows
the structure of a digital communication system with passband transmission via single
AWGN channel, which consists of signal source, sampler, waveform encoder, errorcontrol encoder, digital modulation (digital mapping and analog modulation), AWGN
channel, digital demodulation (analog demodulation and digital demapping), error control
decoder, and lowpass filter. The following subsections describe the above blocks that did
not show up in Section 2, that is, digital modulation and demodulation. Users can display
the required figures at different steps in Figure 30 by clicking the small rectangles, whosecolor stands for a display format specified in the legend at the bottom of Figure 30.
Fig 29 Lowpass filtered signal compared with the original signal
Fig 30 Passband transmission via single AWGN channel
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This section discusses passband digital modulation and demodulation methods.
Digital modulation includes two parts: digital to analog mapping and analog modulation,as shown in Figure 31. Digital to analog mapping techniques convert the received digital
data into analog signals. Analog modulation then modulates the mapped signals.
Typically, digital signals form a finite set of symbols. For example, in binarytransmission it is possible to consider pairs of binary numbers as symbols. In this case,
the symbol set is {00, 01, 10, 11}, making four distinct symbols. A digitalmapping algorithm must have at least four mapping points to map the symbols uniquely.
The number of points in the signal set is called the M-ary number, also known as
alphabet size. The arrangement of the signal set in the signal space, the space that digital
mapping algorithm uses to map the mapping points, is called the constellation.
In general, digital demodulation is the inverse of the modulation process. Amodulated signal is demodulated and then demapped from an analog signal to a digital
signal. Figure 32 shows how digital demodulation divides into an analog demodulationand an analog to digital mapping. This demodulation structure applies to all
demodulation methods except for M-FSK and P-FSK, which use correlation techniques.
3.1 M-ASK
M-ary amplitude shift keying (ASK) modulation includes two parts: M-ASK mapping
followed by analog amplitude modulation (AM). M-ASK mapping is a one-dimensional
coding process that maps the input digital symbols into real numbers in the interval [-x,
x], where x is the specified maximum number. The input symbols are integers in therange [0, M-1], where M is the M-ary number.
Three AM methods are supported in this simulator: double-sideband with
transmission carrier amplitude modulation (AM-DSB-TC), double-sideband carrier
Fig 31 Digital modulation
Fig 32 Digital demodulation
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suppressed amplitude modulation (AM-DSB-SC), and single-sideband amplitude
modulation (AM-SSB). Details on the AM methods can be found in references.
Figure 33 shows the constellation of 4-ASK. Figures 34 through 36 are the spectrums
of the modulated signals using AM-DSB-TC, AM-DSB-SC, and AM-SSB, respectively.
3.2 QASK
The M-ary quadrature amplitude shift keying M-QASK method is the most
commonly used digital modulation method in communication systems. In general, an M-
QASK process takes the input digital symbol and maps it into two independent
components: in-phase and quadrature. An analog quadrature amplitude modulation
(QAM) is used to modulate the in-phase and quadrature signals. On the receiving side,the signal is demodulated into in-phase and quadrature signals. A demapping process
recovers the message signal by using the in-phase and quadrature signals.
There are many mapping schemes available to map the input symbol to the in-phaseand quadrature components. This simulator supports square constellation, as shown in
Figure 37.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
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ASK constellation
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Spectrum of the analog modulated signal
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Fig 33 4-ASK constellation Fig 34 Spectrum using AM-DSB-TC
Fig 35 Spectrum using AM-DSB-SC Fig 36 Spectrum using AM-SSB
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-1.5 -1 -0.5 0 0.5 1 1.5-1.5
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QASK Constallation
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Baseband spectrum for FSK (Hz)
FSK constellation
3.3 M-FSK
M-ary frequency shift keying (FSK) modulation modulates a digital signal bychanging the frequency of the output signal depending on the value of the input signal.
The M-FSK modulation divides into two parts: mapping and analog modulation. The
mapping process maps the input symbol into the value of the frequency shift from the
carrier frequency, and the analog modulation is analog frequency modulation (FM). If the
carrier frequency is Fc, and the tone space f (the frequency separation between twoconsecutive frequencies in the modulated signal), then the frequency range of amodulated signal is in the range [Fc, Fc+(M-1)f]. Figure 38 shows the constellation of4-FSK.
The demodulation process of the M-FSK uses a length-M vector signal where the
frequency of the ith
element in the vector signal matches the modulated signal when the
input symbol is i. The demodulation process computes the correlation values between the
signal array and the received signal and, after calculating the maximum correlation value,
decides what symbol was most likely transmitted. There are two different methods to
compute the correlation value: the coherent and noncoherent method, as shown in Figures
39 and 40, respectively. Using the coherent method, you must know the phaseinformation of the modulated signal from the receiving side where phase-locked loops areused. The noncoherent method does not require phase information; it recovers the phase
of the modulated signal during the demodulation. However, the noncoherentdemodulation method is more computationally complex than the coherent demodulation
method.
3.4 M-PSK
The M-ary phase shift keying (PSK) modulation modulates a signal by changing the
phase values in the modulated output signal. The M-PSK modulation divides into two
parts: mapping and analogphase modulation (PM). M-PSK distinguishes between the
Fig 37 4-ary QASK square constellation Fig 38 4-ary FSK constellation
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digital messages by setting different initial phase shifts in the modulation. The digital
input signals to a M-PSK modulator are in the range [0, M-1]. The phase shift for input
digit i is 2i/M. The structure of the demodulator for M-PSK is similar to that of M-FSK.The demodulation process calculates the correlation value between the input signal and a
vector of carrier frequency sinusoidal signal. Each sinusoidal signal in the vector has its
phase set to a possible result from the signal set. Figure 41 shows the constellation of 4-
PSK. Figure 42 is part of the modulated signal using 4-PSK. We can see phase change in
Figure 42.
4. Passband transmission via multipath fading channel
This section discusses the most important contexts of this report. Figure 43 shows the
ASK/PSK Constel lat ion0 0.2 0.4 0.6 0.8 1 1.2
x 10-5
-1.5
-1
-0.5
0
0. 5
1
1. 5
Analog modulated signal
Fig 39 Coherent demodulation of M-FSK Fig 40 Noncoherent demodulation of M-FSK
Fig 41 Constellation of 4-PSK Fig 42 Modulated signal using 4-PSK
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structure of a digital communication system with passband transmission via multipath
fading channel, which consists of signal source, sampler, waveform encoder, errorcontrol encoder, digital modulation (digital mapping and analog modulation), AWGN
channel, filter bank demodulator, error control decoder, and lowpass filter. The following
subsections discuss the multipath fading channels. Users can display the required figuresat different steps in Figure 43 by clicking the small rectangles, whose color stands for a
display format specified in the legend at the bottom of Figure 43.
4.1 Time variant multipath fading channel
Fundamentally, mobile radio communication channels are time varying, multipath
fading channels. In a radio communication system, there are many paths for a signal totravel from a transmitter to a receiver. Sometimes there is a direct path where the signal
travels without being obstructed. In most cases, components of the signal are reflected by
the ground and objects between the transmitter and the receiver such as buildings,
Fig 43 Digital transmission via multipath fading channel
Fig 44 Multipath fading channel
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vehicles, and hills or refracted by different atmospheric layers. Theses components travel
in different paths and merge at the receiver. Figure 44 illustrates this phenomenon. Eachpath has a different physical length. Thus, signals on each path suffer different
transmission delays due to the finite propagation velocity. The superposition of these
signals at the receiver results in destructive of constructive interference, depending on therelative delays involved. The fact that the environment changes as time passes leads to
signal variation. This is called time variant. Signals are also influenced by the motion of aterminal. A short distance movement can cause an apparent change in the propagation
paths and in turn the strength of the received signals.
4.2 Characterization of multipath fading channel
Both the propagation delays and the attenuation factors are time-variant as a result of
changes in the structure of the medium. The received bandpass signal may be expressedin the form
( ) ( ) ( )( ) =n
nnttsttx (1)
where s(t) is the transmitted signal, ( )tn
is the attenuation factor for the signal received
on the nth
path and ( )tn
is the propagation delay for the nth path. s(t) can be expressed as
( ) ( )[ ]tfjl
cetsts2
Re= (2)
where sl(t) is the equivalent lowpass transmitted signal. Substitute (2) into (1) yields the
result
( ) ( ) ( ) ( )( )
= tfj
n
nl
tfj
ncnc ettsettx
22Re (3)
It is apparent from (3) that the equivalent lowpass received signal is
( ) ( ) ( ) ( )( ) = n
nl
tfj
nl ttsettrnc 2 (4)
It follows that the equivalent lowpass channel is described by the time-variant impulse
response
( ) ( ) ( ) ( )( ) = n
n
tfj
n tettcnc 2; (5)
( )tc ; represents the response of the channel at time tdue to an impulse applied at time t-.
When there are a large number of signal propagation paths, the central limit theorem
can be applied. Thus ( )tc ; can be modeled as a complex-valued Gaussian random
process. The envelope ( )tc ; at any instant tisRayleigh-distributed, as shown in Figure
45. In this case the channel is said to be aRayleigh fading channel. In the event that thereare fixed scatterers or signal reflectors in the medium, in addition to randomly moving
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
1000
2000
3000
4000
5000
6000
0 0.2 0.4 0.6 0.8 1 1.2 1.40
50 0
1000
1500
2000
2500
3000
3500
4000
4500
5000
scatterers, ( )tc ; can no longer be modeled as having zero mean. In this case, the
envelope ( )tc ; has aRice distribution, as shown in Figure 46, and the channel is said to
be aRicean fading channel .
Assuming that c(;t) is wide-sense-stationary (WSS), we can define the
autocorrelation function ofc(;t) as
( ) ( ) ( )[ ]ttctcEtc
+= ;;2
1;, 21
*
21 (6)
where * means conjugation. In most radio transmission media, the attenuation and phase
shift of the channel associated with path delay 1 is uncorrelated with the attenuation and
phase shift associated with path delay 2. This is usually called uncorrelated scattering.Under this assumption, we have
( ) ( ) ( )21121 ;;, = tt cc (7)
( ) ( )0;cc
= is simply the average power output of the channel as a function of the
time delay . For this reason, c() is called the multipath intensity profile or the delay
power spectrum of the channel. The range of values of over which c() is essentiallynonzero is called the multipath spreador delay spreadof the channel and is denoted by
Tm.
The time-variant transfer functionC(f;t) can be defined as the Fourier transform of
c(;t). That is,
( ) ( )
= detctfC fj2;; (8)
If c(;t) is modeled as a complex-valued zero-mean Gaussian random process in the tvariable, it follows that C(f;t) also has the same statistics. Under the assumption that thechannel is WSS, the autocorrelation function of C(f;t) is the Fourier transform of the
multipath intensity profile, i.e.,
( ) ( ) ( ){ }ttfftf cCC == ;;,; 21
(9)
Fig 45 Rayleigh distribution Fig 46 Ricean distribution
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where f=f2-f1. C(f;t) is called the spaced-frequency, spaced-time correlation function
of the channel. Let t=0, then( ) ( ){ }
cCf
= (10)
As a result of the Fourier transform relationship between C(f) and c(), the reciprocalof the multipath spread is a measure of the coherent bandwidth (f)c of the channel. Thatis,
( )m
cT
f1
(11)
If (f)c is small in comparison to the bandwidth of the transmitted signal, the channel is
said to befrequency-selective. In this case, the signal is severely distorted by the channel.On the other hand, if (f)c is large in comparison with the bandwidth of the transmittedsignal, the channel is said to befrequency-nonselective .
Define the Fourier transform ofC(f;t) with respective to the variable tto be thefunction SC(f;). With fset to zero, the relation becomes
( ) ( )
= tdetS tjCC 2;0 (12)
The function SC() is apower spectrum that gives the signal intensity as a function of the
Doppler frequency . Hence, we call SC() the Doppler spectrum of the channel. The
range of values ofover which SC() is essentially nonzero is called the Doppler spreadBdof the channel. Due to the Fourier transform relationship between SC() and C(t), thereciprocal ofBd is a measure of the coherence time (t)c of the channel. That is,
( )d
cB
t1
= (13)
If the coherence time is larger than the symbol period, the channel is said to be a slow-
fading channel. On the other hand, if the coherence time is smaller than the symbol
period, the channel is afast fading channel.
The scattering function of the channel is defined as
( ) ( )
= fdtdeetfS fjtjC 22;; (14)
The relationships among the functions C(f;t), c(;t), C(f;), and s(;) aresummarized in Figure 47.
4.3 Channel model for multipath fading channel
We may view the channel response in (5) as the sum of a number of vectors
(phasors), each of which has a time-variant amplitude n(t) and phase n(t). In general, it
takes large dynamic changes in the physical medium to cause a large change in {n(t)}.On the other hand, the phase {n(t)} change by 2or more radians with relatively small
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changes of the medium characteristics. {n(t)} is expected to be in an unpredictable
(random) manner. Thus c(;t) can be modeled as a complex-valued Gaussian randomprocess using the central limit theorem.
This simulator simulates a slow-fading frequency-selective channel. A general model
for a time-variant slow-fading frequency-selective channel is illustrated in Figure 48. The
channel model consists of a tapped delay line with uniformly spaced taps. The tap
spacing between adjacent taps is 1/W, where W is the bandwidth of the signal transmitted
through the channel. The tap coefficients, denoted as ( ) ( ) ( ){ }tjnn nettc= , are usually
modeled as complex-valued Gaussian random processes that are mutually uncorrelated.
The length of the delay line corresponds to the multipath spread. That is,
W
LT
m= (15)
whereL represents the maximum number of possible multipath signal components. Using
the simulator, the user can specify the means and variances of the possible channels oruse the default values.
Fig 47 Relationships among the channel correlation functions and power spectra
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4.4 Signal design for multipath fading channel
There are lots of carrier modulation methods that are available for digital and analog
modulation, as discussed in Section 3. However, signal fading generally causes large
fluctuations in the received signal amplitude, and consequently it is extremely difficult to
discriminate among multiple amplitude levels in the received signal. For this reason,
ASK and QASK are generally avoided for digital communication through fading
channels. In this simulator, users can choose 2-FSK (coherent), 2-FSK/noncoherent, or 2-
PSK as the digital modulation method. ASK, QASK, and the corresponding analogmodulations are disabled in Figure 43. For simplicity, spread spectrum and ISI are not
discussed in this report.
4.5 RAKE demodulator
Observing Figure 48, it is apparent that the tapped delay line model with statistically
independent tap weights provides us with L replicas of the same transmitted signal at the
receiver. Hence, a receiver that processes the received signal in an optimum manner willachieve the performance of an equivalentL
th-order diversity communications system.
This simulator employs binary signaling over the multipath-fading channel. We have
two equal-energy signals sl1(t) and sl2(t). The received signal is expressed as
( ) ( ) ( ) ( )tzWktstctrL
k
likl += =1
( ) ( ) 2,1,0, =+= iTttztvi
(16)
where z(t) is a complex-valued zero-mean AWGN process. Assume that the channel
tap weights are known. Then the optimum receiver consists of two filters matched to v1(t)
Fig 48 Tapped delay line model of frequency-selective channel
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and v2(t), followed by samplers and a decision circuit that selects the signal
corresponding to the largest output. An equivalent optimum receiver employs cross
correlation instead of matched filtering. In either case, the decision variables for coherentdetection of the binary signals can be expressed as
( ) ( ) 2,1,Re0
* =
= mdttvtrU
T
mlm(17)
Figure 49 illustrates the operations involved in the computation of the decisionvariables. In this realization of the optimum receiver, the two reference signals are
delayed and correlated with the received signal rl(t).
4.6 Simulation results
Figure 50 shows part of the transmitted signal and the corresponding received signal
through the multipath fading channel (without AWGN), whose means and variances of
the tap coefficients are default values shown in Figure 43. Figure 51 shows part of theoriginal binary data and the output of the RAKE demodulator. The bit error rate of this
example is 0.
Fig 49 Optimum demodulator for wideband binary signals
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-6
-1.5
-1
-0.5
0
0.5
1
1.5
Input and Output of the m ult ipath fading channel
Input of the multipath channel
Output of the mult ipath channel
55 60 65 70 75 80 85 90 95 100-1
0
1
2
Original digital signal
55 60 65 70 75 80 85 90 95 100-1
0
1
2
Output of the RAKE demodulater
Reference:
1. J. G. Proakis, Digital Communications, McGraw Hill, 1995;
2. L. W. Couch, Digital and Analog Communication Systems, Prentice Hall,1997;
3. Z. Y. Shen, et al, Principals of Communication Systems, Xidian Univ. Press,1997;
4. J. H. Schiller, Mobile Communications, Addison-wesley, 2000;
5. X. M. Kong, A Simulator for Time-varying Multipath Fading Channels, Master
thesis, 2000.
Fig 50 Input and output of the
multipath-fading channel
Fig 51 Original binary data and the
output of the RAKE demodulator